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Generate Pascal's Triangle to any number of rows.
Pascal's triangle is a triangular array of numbers in which every entry equals the sum of the two entries directly above it. The triangle begins with a 1 at the apex; each row starts and ends with 1, and the interior values build up by that addition rule. This page generates the triangle so you can explore its structure and pull exact values from any row.
The entry in row n, position k (both counted from 0) is the binomial coefficient C(n, k) = n! / (k! (n−k)!). That single connection makes the triangle a compact reference for algebra, probability, and combinatorics: the n-th row gives the coefficients of the expansion of (a + b)^n.
The triangle hides many patterns. The sum of the entries in row n is 2^n, which is why the rows correspond to counting subsets: row n itemizes how many k-element subsets exist in a set of n items. The differences between consecutive entries in the third diagonal give the square numbers, and the shallow diagonals sum to the Fibonacci sequence.
Other well-known features include the symmetry C(n, k) = C(n, n−k), the hockey-stick identity (summing a diagonal returns the entry one step down and across), and the appearance of triangular numbers in the third diagonal. If you shade every odd entry, the triangle reveals the Sierpinski triangle fractal — a classic bridge between simple arithmetic and fractal geometry.
Because row n counts subsets, the triangle solves probability questions such as coin flips and survey sampling. The chance of exactly k heads in n fair flips is C(n, k) / 2^n, so row n of the triangle divided by 2^n gives the full binomial distribution. Row 3 (1, 3, 3, 1), for instance, means three fair flips produce exactly two heads in 3 of 8 equally likely outcomes.
The table below lists the first nine rows. Check the addition rule yourself: each interior number is the sum of the two numbers above it, and each row sums to the next power of two.
| Row n | Entries | Row sum |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 1 | 2 |
| 2 | 1 2 1 | 4 |
| 3 | 1 3 3 1 | 8 |
| 4 | 1 4 6 4 1 | 16 |
| 5 | 1 5 10 10 5 1 | 32 |
| 6 | 1 6 15 20 15 6 1 | 64 |
| 7 | 1 7 21 35 35 21 7 1 | 128 |
| 8 | 1 8 28 56 70 56 28 8 1 | 256 |
To expand (a + b)^n, read the coefficients straight off row n. For example, (a + b)^3 = 1a³ + 3a²b + 3ab² + 1b³, using row 3's entries 1, 3, 3, 1. The exponents of a fall from n to 0 while those of b rise from 0 to n, and the coefficients always sum to 2^n, matching the row sum.
This is the binomial theorem in action: (a + b)^n = Σ C(n, k) a^(n−k) b^k for k from 0 to n. Rather than multiplying out long products by hand, the triangle lets you write down the expansion of (x + 2)^5 directly: x⁵ + 10x⁴ + 40x³ + 80x² + 80x + 32, since each coefficient C(5, k) multiplies the corresponding power of 2.
The triangle appears throughout mathematics precisely because subset counting shows up everywhere: in the binomial distribution of probability, in the number of paths through a grid, in the expansion of polynomials, and in the combinatorics behind the binomial heap and the Sierpinski triangle's self-similar structure. Once you know the rows, you carry a small reference table for all of these in your head.
How do you construct Pascal's triangle?
Start with 1 at the top. Every following row begins and ends with 1, and each interior entry is the sum of the two entries directly above it. Repeat downward for as many rows as needed — no multiplication or factorials are required.
What is the formula for an entry in Pascal's triangle?
The entry at row n, position k (counting both from zero) is the binomial coefficient C(n, k) = n! / (k! (n−k)!). For example, C(5, 2) = 5! / (2! × 3!) = 10, matching the third entry of row 5.
Why does each row of Pascal's triangle sum to a power of 2?
Row n lists how many 0-, 1-, 2-, …, n-element subsets a set of n items has. Summing them counts every possible subset, and a set of n items has exactly 2^n subsets — hence row n sums to 2^n.
How is Pascal's triangle used in probability?
The binomial distribution counts outcomes in repeated yes/no experiments. The probability of exactly k successes in n independent trials with success chance p is C(n, k) p^k (1−p)^(n−k), and the C(n, k) weights come straight from row n of the triangle.
What patterns appear in Pascal's triangle?
Row sums are powers of two, the third diagonal holds triangular numbers, shallow diagonals sum to the Fibonacci sequence, rows are symmetric, and shading the odd entries produces the Sierpinski triangle fractal.
Who discovered Pascal's triangle?
The array was known centuries before Blaise Pascal. It appears in Persian mathematics around the year 1000 in the work of Al-Karaji and Omar Khayyam, in Chinese sources from the 13th century where it is called Yang Hui's triangle, and in Indian writings by Pingala as early as roughly 200 BC, which tabulated combinations of syllables. Pascal's 1654 Traité du triangle arithmétique worked out its properties systematically and connected it to probability, which is why the West attached his name to it.